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The Burnside problem for $\text{Diff}_{\text{Vol}}(\mathbb{S}^2)$
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abstract
Let $S$ be a closed surface and $\text{Diff}_{\text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k \in \mathbb{N}$ such that every element of $G$ has order at most $k$. We show that every periodic group of bounded exponent $G \subset \text{Diff}_{\text{Vol}}(S)$ is a finite group.
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Actions of $2$-groups of bounded exponent on manifolds
Infinite 2-groups of bounded exponent cannot act faithfully by smooth diffeomorphisms on compact manifolds.
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